State estimation for a class of nonlinear time-varying uncertain system under multiharmonic disturbance

TL;DR

The paper tackles state estimation for nonlinear time-varying systems with unknown parameters under a multiharmonic disturbance, assuming a known relative degree $r\le n$ and a regressor linear in the state via $\varphi(x,y,t)=\alpha(y,t)x$ with a harmonic disturbance $f(t)=\sum_{i=1}^q R_i\sin(\omega_i t+\phi_i)$. It proposes a three-stage solution that builds an unknown-input observer, introduces a derivative-free reformulation to relax output-derivative measurements, and then uses autoregression with dynamic regressor extension and mixing (DREM) to recover the unknown parameters $\theta(t)$, the disturbance $f(t)$, and the initial condition vector $\xi(0)$ in finite time. A subsequent time-varying observer is then constructed that uses these estimates to achieve asymptotic reconstruction of the state without requiring output derivatives. The approach is validated by a second-order simulation under measurement noise, illustrating finite-time convergence and robustness, with potential extensions to broader classes of nonlinear time-varying parameter dependencies.

Abstract

The paper considers the observer synthesis for nonlinear, time-varying plants with uncertain parameters under multiharmonic disturbance. It is assumed that the relative degree of the plant is known, the regressor linearly depends on the state vector and may have a nonlinear relationship with the output signal. The proposed solution consists of three steps. Initially, an unknown input state observer is synthesized. This observer, however, necessitates the measurement of output derivatives equal to the plant's relative degree. To relax this limitation, an alternative representation of the observer is introduced. Further, based on this observer, the unknown parameters and disturbances are reconstructed using an autoregression model and the dynamic regressor extension and mixing (DREM) approach. This approach allows the estimates to be obtained in a finite time. Finally, based on these estimates, an observer has been constructed that does not require measurements of the output derivatives. The effectiveness and efficiency of this solution are demonstrated through a computer simulation.

Figures (4)

• Figure 1: The output signal with measurement noise
• Figure 2: The error in estimating the initial conditions of the time-varying parameter $\xi(0)$ and the external disturbance frequency $\omega(t)$
• Figure 3: The error in estimating the external disturbance $f(t)$
• Figure 4: The state observation error $\tilde{x}(t)$